Abstract
We consider a smooth projective morphism between smooth complex projective varieties. If the source space is a weak Fano (or Fano) manifold, then so is the target space. Our proof is Hodge theoretic. We do not need mod p reduction arguments. We also discuss related topics and questions.
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References
Ambro F.: Shokurov’s boundary property. J. Diff. Geom. 67(2), 229–255 (2004)
Ambro F.: The moduli b-divisor of an lc-trivial fibration. Compos. Math. 141(2), 385–403 (2005)
Debarre O.: Higher-dimensional Algebraic Geometry: Universitext. Springer, New York (2001)
Fujino O.: Applications of Kawamata’s positivity theorem. Proc. Jpn. Acad. Ser. A Math. Sci. 75(6), 75–79 (1999)
Fujino O.: A canonical bundle formula for certain algebraic fiber spaces and its applications. Nagoya Math. J. 172, 129–171 (2003)
Fujino, O.: On Kawamata’s theorem, preprint (2007), to appear in the proceedings of the “Classification of Algebraic Varieties” conference, Schiermonnikoog, Netherlands, May 10–15 (2009)
Fujino, O., Gongyo, Y.: On canonical bundle formulae and subadjunctions, preprint (2010). arXiv:1009.3996v1
Hara N., Watanabe K.-i., Yoshida K.: Rees algebras of F-regular type. J. Algebra 247(1), 191–218 (2002)
Hartshorne R.: Algebraic Geometry: Graduate Texts in Mathematics, No. 52. Springer, New York (1977)
Kawamata, Y.: Subadjunction of log canonical divisors for a subvariety of codimension 2, Birational algebraic geometry (Baltimore, MD, 1996), 79–88, Contemp. Math., 207: Am. Math. Soc., Providence, RI (1997)
Kawamata Y.: Subadjunction of log canonical divisors. II. Am. J. Math. 120(5), 893–899 (1998)
Kollár J., Miyaoka Y., Mori S.: Rational connectedness and boundedness of Fano manifolds. J. Diff. Geom. 36(3), 765–779 (1992)
Kollár, J., Mori, S.: Birational geometry of algebraic varieties. With the collaboration of C. H. Clemens and A. Corti. Translated from the 1998 Japanese original. Cambridge Tracts in Mathematics, 134. Cambridge University Press, Cambridge (1998)
Liu Q.: Algebraic Geometry and Arithmetic Curves, Translated from the French by Reinie Ernè. Oxford Graduate Texts in Mathematics, 6. Oxford Science Publications, Oxford University Press, Oxford (2002)
Miyaoka Y.: Relative deformations of morphisms and applications to fibre spaces. Comment. Math. Univ. St. Paul. 42(1), 1–7 (1993)
Prokhorov Yu. G., Shokurov V.V.: Towards the second main theorem on complements. J. Algebr. Geom. 18(1), 151–199 (2009)
Schwede K., Smith K.E.: Globally F-regular and log Fano varieties. Adv. Math. 224(3), 863–894 (2010)
Smith K.E.: Globally F-regular varieties: applications to vanishing theorems for quotients of Fano varieties, Dedicated to William Fulton on the occasion of his 60th birthday. Mich. Math. J. 48, 553–572 (2000)
Wiśniewski J.A.: On contractions of extremal rays of Fano manifolds. J. Reine Angew. Math. 417, 141–157 (1991)
Yasutake, K.: On the classification of rank 2 almost Fano bundles on projective space. preprint (2010). arXiv:1004.2544v2
Zhang Qi.: On projective manifolds with nef anticanonical bundles. J. Reine Angew. Math. 478, 57–60 (1996)
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Fujino, O., Gongyo, Y. On images of weak Fano manifolds. Math. Z. 270, 531–544 (2012). https://doi.org/10.1007/s00209-010-0810-6
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DOI: https://doi.org/10.1007/s00209-010-0810-6